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PHYSICAL REVIEW B 74, 054202 共2006兲
Complexity in charge transport for multiwalled carbon nanotube
and poly(methyl methacrylate) composites
Heon Mo Kim, Mahn-Soo Choi, and Jinsoo Joo*
Department of Physics and Institute for Nano Science, Korea University, Seoul 136-713, Korea
Sin Je Cho
Iljin Nanotech Co., Ltd., Kangseo-Ku, Seoul 157-810, Korea
Ho Sang Yoon
NOVATEMS Inc., DangJeong-Dong, Gunpo-City, Kyunggi-Do 435-831, Korea
共Received 10 April 2006; revised manuscript received 22 June 2006; published 3 August 2006兲
We report on studies of the charge transport mechanism of the composites of multiwalled carbon nanotubes
共MWCNTs兲 and poly 共methyl methacrylate兲 共PMMA兲. The MWCNTs were synthesized by a chemical vapor
deposition method using Fe as a catalyst. The dc conductivity 共␴dc兲 and its temperature dependence 关␴dc共T兲兴
were measured in a temperature range of 0.5– 300 K to study the charge transport mechanism of the composites. The ␴dc of composites at room temperature increased as the MWCNT concentration increased, which
shows typical percolation behavior with percolation threshold of pc at ⬃0.4 wt % of the MWCNTs. The ␴dc共T兲
of the MWCNT-PMMA composites were fitted to the combination of Sheng’s fluctuation induced tunneling
共FIT兲 model and the one-dimensional variable range hopping 共1D VRH兲 model. The tunneling mechanism and
the 1D VRH transport were attributed to the charge tunneling between MWCNT clusters and the charge
hopping through 1D MWCNTs, respectively. Magnetoresistance 共MR兲 of MWCNT-PMMA composites was
measured up to 9 T of magnetic field. The results of MR and ␴dc共T兲 showed that the FIT model was dominant
in the low temperature region 共T ⱗ 10 K兲, while the 1D VRH process became dominant as the temperature
increased 共10 K ⱗ T ⱗ 300 K兲. We observed unusually large negative MR in the composites at the lower
temperatures 共T ⬍ 4 K兲 due to FIT conduction. We related the parameters specifying charge transport to the
percolation structures of MWCNT-PMMA composites based on the results of temperature dependence of ␴dc
and of the MR.
DOI: 10.1103/PhysRevB.74.054202
PACS number共s兲: 73.63.Fg, 72.15.Cz, 71.20.Lp, 73.43.Qt
I. INTRODUCTION
Research on carbon nanotubes has rapidly increased during this last decade. Single-walled carbon nanotubes
共SWCNTs兲 have been studied in the academic field because
of their low dimensional nature with well-defined structure.1
Multiwalled carbon nanotubes 共MWCNTs兲 have been regarded as excellent nanomaterials for fundamental research
as well as industrial applications.2,3 Research is important
not only for the application of MWCNTs and not only for
understanding intrinsic properties of MWCNTs, but also for
low cost mass production of these MWCNTs. The composites of MWCNTs with conventional polymers have been emphasized for commercial applications. In these applications,
the electrical and thermal properties of the MWCNTs should
be optimized with flexibility and an easy mixing process
obtained from the polymers. To optimize the electrical properties in the composites, it is required to study the charge
transport mechanism for the various composites of the
MWCNTs with insulating polymers.4,5 Such studies include
the temperature dependence of electrical conductivity and of
magnetoresistance 共MR兲.
The electrical conduction in the composites of conducting
MWCNTs and insulating poly 共methyl methacrylate兲
共PMMA兲 have been known as percolating behavior.
The studies of MWCNT composites with poly 共 mphenylenevinylene-co-2,5-dioctyloxy-p-phenylenevinylene 兲
1098-0121/2006/74共5兲/054202共7兲
or polyvinylalcohol showed that the charge transport mechanism was mainly affected by fluctuation induced tunneling
共FIT兲. Here the dc conductivity 共␴dc兲 at room temperature
共RT兲 关␴dc共RT兲兴 and the ac conductivity 共␴ac兲 as a function of
MWCNT concentration had been measured for the analysis.5
Sheng’s FIT model6 has been in competition with the variable range hopping 共VRH兲 model7 in explaining the charge
transport mechanism in the composites of MWCNTs and
PMMA. In order to clarify the charge transport mechanism
of the MWCNT-polymer composites, experiments addressing the temperature dependence of ␴dc 关␴dc共T兲兴 and the MR
in the lower temperature region 共T ⱗ 4 K兲 should be performed and results then analyzed.
In this study, we synthesized composites of MWCNTs and
PMMA in the form of free-standing films. The percolation
behavior as a function of the mass fraction of the MWCNTs
in the PMMA background was observed from the measured
␴dc共RT兲. From ␴dc共T兲 in the temperature range from
0.5 to 300 K, the combination of Sheng’s FIT model and the
one-dimensional 共1D兲 VRH model was suggested for explaining the charge transport mechanism of the systems. The
results of temperature and magnetic field dependence of
the MR of the systems supported the complexity of charge
transport using the combination of the FIT and 1D VRH
models.
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©2006 The American Physical Society
PHYSICAL REVIEW B 74, 054202 共2006兲
KIM et al.
FIG. 1. 共a兲 dc conductivity at
room temperature 关␴dc共RT兲兴 of
MWCNT-PMMA composites as
a function of the MWCNT mass
concentrations 共p兲. Inset: Plot of
␴dc共RT兲 as a function of p−1/3.
共b兲 Temperature dependence of dc
conductivity
关␴dc共T兲兴
of
MWCNT-PMMA composites.
II. EXPERIMENT
The MWCNTs were synthesized by the chemical vapor
deposition method using Fe as the catalyst, which were provided by Iljin Nanotech Co., Ltd.8 The MWCNTs were dispersed in toluene with PMMA through stirring and sonication for 24 h. The weight 共wt兲% of MWCNTs varied from
0.1 to 40 wt %. The dispersed solution was cast onto a teflon
substrate and dried in a vacuum oven at 40 ° C over 12 h to
prepare free-standing films. The thickness of the freestanding films was from ⬃60 to ⬃ 165 ␮m. The homogeneous dispersion of MWCNTs in the composites was confirmed by means of SEM, TEM, and XPS, which were
reported earlier.3,9
For the measurement of ␴dc共T兲, the 4-probe contact
method was used to eliminate contact resistance. The mea-
surement for ␴dc共T兲 was in the temperature range from
0.5 to 300 K and was made by using the physical property
measurement system 共Quantum Design 6000 PPMS兲. Quantum design model 6000 9T PPMS was used for measuring
the MR in the temperature range of 0.5 to 300 K with the
magnetic field up to 9 T. The applied magnetic field for the
MR measurements of the systems was perpendicular to the
direction of the applied current in the 4-probe contact
method.
III. RESULTS AND DISCUSSION
Figure 1共a兲 shows ␴dc共RT兲 as a function of the MWCNT
mass concentrations 共p兲. The ␴dc共RT兲 of the composites increased up to ten orders in magnitudes as the MWCNT con-
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COMPLEXITY IN CHARGE TRANSPORT FOR¼
FIG. 2. Temperature dependence of resistivity of MWCNT-PMMA composites with 共a兲 0.4 wt %, 共b兲 2 wt %, 共c兲 5 wt %, and 共d兲
10 wt % of MWCNT concentration. The fitting lines were from the linear combination of the Sheng’s fluctuation induced tunneling 共FIT兲
model and the one-dimensional variable range hopping 共1D VRH兲 model.
centration increased up to 30 wt %. The plot of ␴dc共RT兲 vs p
for the MWCNT-PMMA composites showed typical percolation behavior described as ␴共p兲 ⬀ 兩p − pc兩t. The critical exponent t in Fig. 1共a兲 varied from 11± 4 to 2.3± 0.4 at
⬃0.4 wt % of MWCNT concentration. This implies that the
percolation threshold was at a pc ⬃ 0.004. The pc was converted to the volume fraction as f c ⬃ 0.0087. For conventional three-dimensional 共3D兲 systems, the percolation
threshold by volume fraction is 0.16.10 The relatively small
value of the percolation threshold in the MWCNT composites studied here was accounted for MWCNTs having onedimensional 共1D兲 shape with a large aspect ratio. The inset of
Fig. 1共a兲 shows the plot of ␴dc共RT兲 vs p−1/3, which followed
the expected linear relation. Kilbride et al. reported the
charge transport mechanism of the percolating system consisted of CNTs and insulating polymers. It was described by
the FIT model,5 where the relation between ␴dc and the
MWCNT concentration 共p兲 was described as ln ␴dc ⬀ −p−1/3.
However, the deviation from the fitting line at p ⬍ 0.004 as
shown in the inset of Fig. 1共a兲 implies that another charge
transport process might be considered in the composites. Fig-
ure 1共b兲 shows ␴dc共T兲 in the MWCNT-PMMA composites
above the percolation threshold 共pc = 0.004兲. The ␴dc of all
composites of MWCNT-PMMA decreased with decreasing
temperature, implying charge localization. The ratios of
␴dc共RT兲 to ␴dc共T = 2.1 K兲 关␴r ⬅ ␴dc共RT兲 / ␴dc共T = 2.1 K兲兴
TABLE I. Fitting parameters of T1, T0, and T1D for the
MWCNT-PMMA composites based on the combination of the
Sheng’s FIT and the 1D VRH models.
wt % of
MWCNTs
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0.4
1
2
5
10
20
30
T1
共K兲
T0
共K兲
T1D
共K兲
195.13
39.23
29.80
13.90
8.21
8.13
8.92
6.68
1.58
1.02
0.55
0.48
0.49
0.66
108.99
77.55
62.69
31.31
11.31
11.55
9.80
PHYSICAL REVIEW B 74, 054202 共2006兲
KIM et al.
FIG. 3. 共a兲 Variation of percolation exponent t
at different temperatures. Inset table: Values of t
at different temperatures. The dotted lines are a
guide to the eye. 共b兲 Temperature dependence of
percolation critical exponent t above 10 K based
1/2
on the 1D VRH model. Inset: T1D
vs −ln共p − pc兲;
percolation behavior of activation energy 共T1D兲 of
the 1D VRH model.
were calculated. For the composites with MWCNT concentrations of 10– 30 wt %, temperature dependences of ␴r were
relatively weak as ␴r = 8.40– 11.81. However, for the composites with MWCNT concentrations below 2 wt %, ␴r increased considerably up to ⬃1.9⫻ 104 as the MWCNT concentration decreased, indicating more insulating behavior.
Figures 2共a兲–2共d兲 show plots of dc resistivity 共␳dc兲 vs T −1
for the composite films of 0.4, 2, 5, and 10 wt % of MWCNT
concentration, respectively. We analyzed that it was impossible to fit the temperature dependence of resistivity 关␳dc共T兲
⬅ ␴dc−1共T兲兴 over the whole temperature region including the
lower temperature range 共0.5 K 艋 T 艋 300 K兲 just by using
Sheng’s FIT model. Because of the complexity of charge
transport in MWCNT-PMMA composites, we used a linear
combination of Sheng’s FIT model and the 1D VRH model
as follows:11,12
冋冉 冊册
␴dc共T兲 = ␴1D exp −
T1D
T
1/2
冉 冊
+ ␴0 exp
− T1
,
T + T0
共1兲
where the first and the second terms originated from the 1D
VRH model and the Sheng’s FIT model, respectively.6,7
Here, ␴1D and ␴0 are the proportionality constants, T1D is the
activation energy in the 1D VRH model, T1 is the temperature below which the conduction is dominated by the charge
tunneling through the barrier, and T0 is the temperature
above which the thermally activated conduction over the barrier begins to occur. The ␳dc共T兲’s of all composites of
MWCNT-PMMA studied here were fitted to the Eq. 共1兲, as
shown in Fig. 2. From the results of ␳dc共T兲 and the fitting
curves based on Eq. 共1兲, Sheng’s FIT model was dominant
for the charge transport mechanism in composites below
10 K, which can be attributed to tunneling conduction
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COMPLEXITY IN CHARGE TRANSPORT FOR¼
T1D ⬀ 关ln共p − pc兲兴2 .
共3兲
For n-dimensional conductors dominated by the VRH conduction, Eq. 共3兲 can be generalized to be
Tn−D ⬀ 关ln共p − pc兲兴n+1 .
共4兲
Below 10 K, the charge transport in MWCNT-PMMA composites was dominated by the Sheng’s FIT process. The percolation exponent t due to the tunneling conduction was related to the second fitting term of Eq. 共1兲 with ␴ = ␴0⬘共p
− pc兲t. We, then, obtained the following relation:
t=
FIG. 4. Temperature dependence 共0.7 K 艋 T 艋 300 K兲 of the
MR of MWCNT-PMMA composites with 20 and 30 wt % of
MWCNTs at 9 T of magnetic field. Inset: Temperature dependence
of absolute value of MR in log scale.
冉
冊
ln共␴0/␴0⬘兲
−1
T 1 T 0T 1
− 2 +
.
ln共p − pc兲 T
T
ln共p − pc兲
共5兲
This is not a simple equation to obtain the generalized relation of T0 and T1 with p − pc.
Figure 4 presents temperature dependence of MR
关MR共T兲兴 of MWCNT-PMMA composites with MWCNT
concentrations of 20 and 30 wt % from 0.7 to 300 K. As the
temperature decreased, the MR of MWCNT-PMMA composites negatively increased. The inset of Fig. 4 shows absolute
through insulating PMMA barriers between MWCNT networks. As temperature increased above 10 K, the 1D VRH
process between MWCNTs or MWCNT networks became
dominant as shown in Figs. 2共a兲–2共d兲. It is noted that the
␳dc共T兲 of the composites with 20 and 30 wt % of MWCNTs
was also fitted to Eq. 共1兲. The values of T1, T0, and T1D,
obtained from the fitting curves of the FIT and 1D VRH
models are listed in Table I. The values of T1, T0, and T1D
decreased with increasing concentration of MWCNTs, implying the lowering of activation energies.
Figure 3共a兲 shows the variation of percolation exponent t
at different temperatures obtained from the results of ␴dc共T兲
and ␴ = ␴0⬘共p − pc兲t. The percolation exponent t increased as
temperature decreased as listed in the inset table of Fig. 3共a兲.
Our analysis indicated that the percolation behavior of
MWCNT-PMMA composites was affected by temperature
due to the different activation energies. In general, the percolation exponent t increases as the dimensionality of the
systems increases, and pc of the system also depends on the
dimensionality of the systems.10 When the percolation relation ␴ = ␴0⬘共p − pc兲t is combined with the first term of Eq. 共1兲,
i.e., the 1D VRH model, the relation between the t and temperature above 10 K can be described as in Eq. 共2兲:
t=
冉 冊
−1
T1D
ln共p − pc兲 T
1/2
+
ln共␴1D/␴0⬘兲
⬀ T −1/2 .
ln共p − pc兲
共2兲
Figure 3共b兲 shows t vs T −1/2 based on Eq. 共2兲. The values of
t above 10 K were fitted to Eq. 共2兲 with the slope of ⬃1.63.
The activation energy T1D was directly related to ln共p − pc兲
through the value of the slope. The inset of Fig. 3共b兲 shows
1/2
as a function of −ln共p − pc兲, using the T1D values in Table
T1D
1/2
I. The slope of the plot of T1D
vs −ln共p − pc兲 was ⬃1.75,
which was similar to the value of the slope 共⬃1.63兲 using
Eq. 共2兲. Therefore the activation energy in the 1D VRH
model can be described as a function of the MWCNT concentration 共p − pc兲,
FIG. 5. 共a兲 MR vs H2 at various temperatures for 20 wt % of
MWCNT-PMMA composites. 共b兲 MR vs H2 at various temperatures for 30 wt % of MWCNT-PMMA composites.
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KIM et al.
values of MR共T兲 of the systems on a log scale. From 300 K
down to ⬃10 K, we observed that the MR negatively increased by ⬃3% compared to the zero field resistance
关R共H = 0兲兴. As the temperature decreased from 10 to 0.7 K,
the negative MR of the MWCNT-PMMA composites rapidly
increased from ⬃2.8% to ⬃32% compared to R共H = 0兲, as
shown in the inset of Fig. 4. This suggests that a different
transport mechanism might be applied in the lower temperature range. The rapid change of the negative MR below
⬃10 K qualitatively agrees with the results and analysis of
␴dc共T兲.
The negative MR can be caused by the VRH process, FIT,
or weak localization effect.12,13 For the networks of
SWCNTs or MWCNTs, the results of the MR have been
explained by the VRH model12 or the weak localization
effect.13 The negative MR in terms of the VRH model is
related to the applied magnetic field 共H兲 and temperature
described as
冉 冊冋
␴共H,T兲 − ␴共0,T兲 1 ␤n
⬀
2 n+1
␴共0,T兲
2
g ␮ BH
2共EC − EF兲
册冉 冊
2
T0
T
2/共n+1兲
,
共6兲
where 兩 21 g␮BH / 共EC − EF兲兩 Ⰶ 1 is assumed, i.e., the weak magnetic field limit. Here, ␤ is approximately 1, n is the dimensionality constant, g is the effective g-factor, ␮B is the Bohr
magneton, and EC is the mobility edge.14 Figures 5共a兲 and
5共b兲 show magnetic field dependence of the MR of
MWCNT-PMMA composites, with 20 and 30 wt % of
MWCNTs, respectively, at various temperatures 共5 K 艋 T
艋 100 K兲. Based upon Eq. 共6兲, it is expected that the MR
would show the quadratic magnetic field dependence 共⬀H2兲
in the weak field limit, which was determined from 21 kBT
⬇ ␮BH 艋 兩EC − EF兩. With T 艌 30 K, the MR of the composites
with 20 and 30 wt % of MWCNTs followed the quadratic
field dependence in the weak field limit, which supports the
domination of the VRH model for the charge transport
mechanism. As shown in Figs. 5共a兲 and 5共b兲, the MR in the
*Corresponding author. FAX: ⫹82-2-927-3292; Email address:
[email protected]
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field dependence in the whole applied magnetic field. This
might suggest that another transport model, such as the FIT
model, can dominate the charge transport mechanism in
lower temperature range. The results of the magnetic field
dependence of the MR also supported the results and analysis of ␴dc.
IV. SUMMARY
We synthesized MWCNT-PMMA composites with various concentrations of MWCNTs. The measured ␴dc共RT兲 of
the MWCNT-PMMA composites showed percolation behavior with relatively lower percolation threshold 共pc ⬃ 0.004兲,
which originated from the 1D shape of conducting MWCNTs
and their network formation in the composites. The charge
transport properties of the MWCNT-PMMA composites
were explained in terms of a combination of the Sheng’s FIT
and the 1D VRH models. The relation between the parameters specifying charge transport and percolation behavior
were described as Tn−D ⬀ 关ln共p − pc兲兴n+1 for the n-dimensional
VRH model. From the measurements of ␳dc共T兲 and the MR,
we observed that the 1D VRH conduction was dominant in
charge transport at a relatively higher temperature 共10 K
ⱗ T ⱗ 300 K兲 for the composites with higher concentrations
of MWCNTs. The Sheng’s FIT model was fitted to the results of ␳dc共T兲 at a relatively lower temperature region 共T
ⱗ 10 K兲. The results of the temperature and magnetic field
dependences of the MR supported the results of ␴dc. As the
temperature decreased to 0.7 K, the negative MR unusually
increased up to ⬃30% compared to zero field resistance.
ACKNOWLEDGMENTS
This work was supported by a Korean Research Foundation Grant 共KRF-2004-005-C00068兲 and the SRC 共R112000-071兲. The authors thank S. H. Park at Korean Basic
Science Institute in Daejeon for assisting measurements of
dc conductivity and magnetoresistance.
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